Fungrim entry: cb5071

\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma\!\left(x\right)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

References:

L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5))

TeX:

\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma\!\left(x\right)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Factorial`	$n !$	Factorial
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("cb5071"),
    Formula(LessEqual(Abs(Mul(Div(1, Factorial(n)), Derivative(Div(1, GammaFunction(x)), Tuple(x, 0, n)))), Div(2, Sqrt(Factorial(n))))),
    Variables(n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)))),
    References("L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5))"))

Topics using this entry

Bounds and inequalities for the gamma function